
R version 3.5.3 (2019-03-11) -- "Great Truth"
Copyright (C) 2019 The R Foundation for Statistical Computing
Platform: x86_64-w64-mingw32/x64 (64-bit)

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Microsoft R Open 3.5.3
The enhanced R distribution from Microsoft
Microsoft packages Copyright (C) 2019 Microsoft Corporation

Using the Intel MKL for parallel mathematical computing (using 18 cores).

Default CRAN mirror snapshot taken on 2019-04-15.
See: https://mran.microsoft.com/.

[Previously saved workspace restored]

> #################################
> # Figure 1
> #################################
> 
> rm(list=ls())
> 
> library(copula)
> library(ggplot2)
> library(ggExtra)
> 
> set.seed(123456)
> 
> N <- 3000
> norm.cop <- normalCopula(0)
> draws1 <- data.frame( 100*rCopula(N, norm.cop) )
> names(draws1) <- c("SET Score Percentile", "Instructor Quality Percentile")
> 
> p <- ggplot(draws1, aes(`SET Score Percentile`, `Instructor Quality Percentile`)) + geom_point(color="darkgray", alpha=0.25) + theme_classic() + theme(panel.border = element_rect(fill=NA))
> q <- ggMarginal(p, type = "histogram", fill="gray", xparams = list(size=0.2), yparams = list(size=0.2))
> ggsave(filename="correlation-zero.pdf", plot=q, width=3, height=(2.8 / 5.78) * 6.04, units="in")
> 
> 
> 
> 
> rm(list=ls())
> N <- 3000
> norm.cop <- normalCopula(0.4)
> draws1 <- data.frame( 100*rCopula(N, norm.cop) )
> names(draws1) <- c("SET Score Percentile", "Instructor Quality Percentile")
> 
> p <- ggplot(draws1, aes(`SET Score Percentile`, `Instructor Quality Percentile`)) + geom_point(color="darkgray", alpha=0.25) + theme_classic() + theme(panel.border = element_rect(fill=NA))
> q <- ggMarginal(p, type = "histogram", fill="gray", xparams = list(size=0.2), yparams = list(size=0.2))
> ggsave(filename="correlation-mod.pdf", plot=q, width=3, height=(2.8 / 5.78) * 6.04, units="in")
> 
> 
> 
> 
> rm(list=ls())
> N <- 3000
> norm.cop <- normalCopula(0.9)
> draws1 <- data.frame( 100*rCopula(N, norm.cop) )
> names(draws1) <- c("SET Score Percentile", "Instructor Quality Percentile")
> 
> p <- ggplot(draws1, aes(`SET Score Percentile`, `Instructor Quality Percentile`)) + geom_point(color="darkgray", alpha=0.25) + theme_classic() + theme(panel.border = element_rect(fill=NA))
> q <- ggMarginal(p, type = "histogram", fill="gray", xparams = list(size=0.2), yparams = list(size=0.2))
> ggsave(filename="correlation-high.pdf", plot=q, width=3, height=(2.8 / 5.78) * 6.04, units="in")
> 
> 
> #################################
> # Appendix Figure 8
> #################################
> 
> rm(list=ls())
> 
> library(mvtnorm)
> library(ggplot2)
> library(ggExtra)
> 
> set.seed(123456)
> 
> N <- 3000
> Sigma = matrix(data=c(1, 0, 0, 1), nrow=2, ncol=2)
> draws1 <- data.frame( rmvnorm(N, mean=c(0,0), sigma=Sigma) )
> names(draws1) <- c("SET Score", "Instructor Quality Metric")
> 
> p <- ggplot(draws1, aes(`SET Score`, `Instructor Quality Metric`)) + geom_point(color="darkgray", alpha=0.25) + theme_classic() + theme(panel.border = element_rect(fill=NA))
> q <- ggMarginal(p, type = "histogram", fill="gray", xparams = list(size=0.2), yparams = list(size=0.2))
> ggsave(filename="correlation-normal-zero.pdf", plot=q, width=3, height=(2.8 / 5.78) * 6.04, units="in")
> 
> 
> 
> 
> rm(list=ls())
> N <- 3000
> Sigma = matrix(data=c(1, 0.4, 0.4, 1), nrow=2, ncol=2)
> draws1 <- data.frame( rmvnorm(N, mean=c(0,0), sigma=Sigma) )
> names(draws1) <- c("SET Score", "Instructor Quality Metric")
> 
> p <- ggplot(draws1, aes(`SET Score`, `Instructor Quality Metric`)) + geom_point(color="darkgray", alpha=0.25) + theme_classic() + theme(panel.border = element_rect(fill=NA))
> q <- ggMarginal(p, type = "histogram", fill="gray", xparams = list(size=0.2), yparams = list(size=0.2))
> ggsave(filename="correlation-normal-mod.pdf", plot=q, width=3, height=(2.8 / 5.78) * 6.04, units="in")
> 
> 
> 
> 
> rm(list=ls())
> N <- 3000
> Sigma = matrix(data=c(1, 0.9, 0.9, 1), nrow=2, ncol=2)
> draws1 <- data.frame( rmvnorm(N, mean=c(0,0), sigma=Sigma) )
> names(draws1) <- c("SET Score", "Instructor Quality Metric")
> 
> 
> p <- ggplot(draws1, aes(`SET Score`, `Instructor Quality Metric`)) + geom_point(color="darkgray", alpha=0.25) + theme_classic() + theme(panel.border = element_rect(fill=NA))
> q <- ggMarginal(p, type = "histogram", fill="gray", xparams = list(size=0.2), yparams = list(size=0.2))
> ggsave(filename="correlation-normal-high.pdf", plot=q, width=3, height=(2.8 / 5.78) * 6.04, units="in")
> 
> 
> 
> 
> #################################
> # Appendix Figure 9
> #################################
> 
> rm(list=ls())
> 
> library(mvtnorm)
> set.seed(123456)
> 
> N <- 500000
> rho <- seq(0.01, 0.99, 0.01)
> error.rate <- c()
> improvement <- c()
> 
> pb <- txtProgressBar(0, length(rho), style=3)
  |                                                                              |                                                                      |   0%> for(i in seq_along(rho)){
+   
+   setTxtProgressBar(pb, i
+                     )
+   sigma.mat <- matrix(data=c(1, rho[i], rho[i], 1), nrow=2, ncol=2, byrow=T)
+   draws1 <- rmvnorm(N, mean=c(0,0), sigma=sigma.mat)
+   draws2 <- rmvnorm(N, mean=c(0,0), sigma=sigma.mat)
+   
+   #eval.m <- draws1[,1] > draws2[,1]
+   #effect.m <- draws1[,2] > draws2[,2]
+   correct1 <- (draws1[,1] > draws2[,1]) & (draws1[,2] > draws2[,2])
+   correct2 <- (draws1[,1] < draws2[,1]) & (draws1[,2] < draws2[,2])
+   correct3 <- (draws1[,1] == draws2[,1]) & (draws1[,2] == draws2[,2])
+   
+   correct <- correct1 + correct2 + correct3
+   error.rate[i] <- 100*(1 - sum(correct)/N)
+   improvement[i] <- (sum(correct)/N - 0.5)/0.5
+   
+ }
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|======================================================================| 100%> close(pb)

> 
> pdf(file="error-rate.pdf", width=5, height=5.5)
> 
> par(mar=c(5,5,4,2))
> 
> scatter.smooth(error.rate ~ rho,
+                xlab = c("correlation between true instructor quality", "and average student evaluation"),
+                ylab = "", #"error rate (% of time higher quality teacher has lower average evaluation)",
+                span = 0.2, degree=2, col="darkgray"
+ )
> 
> mtext(text = "error rate (% of time higher quality instructor", side=2, line=4)
> mtext(text = "has lower average evaluation)", side=2, line=3)
> 
> clip(-1,.4,-1,error.rate[rho==0.4])
> abline(v=0.4, col="darkgray", lty=2)
> abline(h=error.rate[rho==0.4], col="darkgray", lty=2)
> clip(-1,0.9,-1,error.rate[rho==0.9])
> abline(v=0.9, col="darkgray", lty=2)
> abline(h=error.rate[rho==0.9], col="darkgray", lty=2)
> box()
> 
> 
> dev.off()
pdf 
  2 
> 
> 
> #################################
> # Appendix Figure 10
> #################################
> 
> 
> rm(list=ls())
> 
> library(mvtnorm)
> set.seed(123456)
> 
> N <- 500000
> rho <- 0.4
> gap <- seq(from=0, to=2, by=0.01)
> error.rate <- c()
> decision.rate <- c()
> 
> pb <- txtProgressBar(0, length(gap), style=3)
  |                                                                              |                                                                      |   0%> for(i in seq_along(gap)){
+   
+   setTxtProgressBar(pb, i)
+   sigma.mat <- matrix(data=c(1, rho, rho, 1), nrow=2, ncol=2, byrow=T)
+   draws1 <- rmvnorm(N, mean=c(0,0), sigma=sigma.mat)
+   draws2 <- rmvnorm(N, mean=c(0,0), sigma=sigma.mat)
+   
+   eval.m <- ifelse(draws1[,1] - draws2[,1] > gap[i], 1,
+                    ifelse( draws2[,1] - draws1[,1] > gap[i], -1, 0))
+   #effect.m <- draws1[,2] > draws2[,2]
+   
+   decision <- eval.m != 0
+   incorrect <- (eval.m == 1 & (draws1[,2] < draws2[,2])) | (eval.m == -1 & (draws1[,2] > draws2[,2]))
+   error.rate[i] <- 100*(sum(incorrect)/N)
+   decision.rate[i] <- 100*(sum(decision)/N)
+ 
+ }
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> close(pb)

> 
> pdf(file="gap-error.pdf", width=5.5, height=5.85)
> 
> par(mar=c(5,5,4,2))
> scatter.smooth(error.rate ~ gap,
+                xlab = c("minimum diff. in average evaluation scores","(in percentiles) required for decision"),
+                ylab = "",
+                span = 0.2, degree=2, col="darkgray", ylim=c(0,40)
+ )
> text(1.15, 40, labels = paste("correlation between avg. evaluation and quality = ", rho, sep=""),
+      cex=0.8)
> mtext(text = "error rate (% of time better teacher has average evaluation",
+       side=2, line=4)
> mtext(text = "lower by more than the percentile gap on the x-axis)",
+       side=2, line=3)
> 
> x1 <- gap[which(abs(10-error.rate) == min(abs(10-error.rate)))]
> x2 <- gap[which(abs(5-error.rate) == min(abs(5-error.rate)))]
> clip(-1,x1,-10,10)
> abline(v=x1, col="darkgray", lty=2)
> abline(h=10, col="darkgray", lty=2)
> clip(-1,x2,-10,5)
> abline(v=x2, col="darkgray", lty=2)
> abline(h=5, col="darkgray", lty=2)
> box()
> 
> dev.off()
pdf 
  2 
> 
> gap[which(abs(10-error.rate) == min(abs(10-error.rate)))]
[1] 1.27
> error.rate[which(abs(10-error.rate) == min(abs(10-error.rate)))]
[1] 9.9338
> 
> gap[which(abs(5-error.rate) == min(abs(5-error.rate)))]
[1] 1.75
> error.rate[which(abs(5-error.rate) == min(abs(5-error.rate)))]
[1] 4.9954
> 
> pdf(file="gap-decision.pdf", width=5, height=5.5)
> 
> par(mar=c(5,5,4,2))
> scatter.smooth(decision.rate ~ gap,
+                xlab = "diff. in average evaluation scores (in standard deviations)",
+                ylab = "",
+                span = 0.2, degree=2, col="darkgray"
+ )
> text(1.4, 100, labels = paste("correlation between avg. evaluation and quality = ", rho, sep=""),
+      cex=0.8)
> mtext(text = "decision rate (% of time teacher has average evaluation",
+       side=2, line=4)
> mtext(text = "that is lower by number of points indicated on x-axis)",
+       side=2, line=3)
> 
> dev.off()
pdf 
  2 
> 
> 
> 
> 
> #################################
> # Figure 2
> #################################
> 
> 
> rm(list=ls())
> 
> library(copula)
> set.seed(123456)
> 
> N <- 500000
> rho <- seq(0.01, 0.99, 0.01)
> error.rate <- c()
> improvement <- c()
> 
> pb <- txtProgressBar(0, length(rho), style=3)
  |                                                                              |                                                                      |   0%> for(i in seq_along(rho)){
+   
+   setTxtProgressBar(pb, i)
+   norm.cop <- normalCopula(rho[i])
+   draws1 <- rCopula(N, norm.cop)
+   draws2 <- rCopula(N, norm.cop)
+   
+   #eval.m <- draws1[,1] > draws2[,1]
+   #effect.m <- draws1[,2] > draws2[,2]
+   correct1 <- (draws1[,1] > draws2[,1]) & (draws1[,2] > draws2[,2])
+   correct2 <- (draws1[,1] < draws2[,1]) & (draws1[,2] < draws2[,2])
+   correct3 <- (draws1[,1] == draws2[,1]) & (draws1[,2] == draws2[,2])
+   
+   correct <- correct1 + correct2 + correct3
+   error.rate[i] <- 100*(1 - sum(correct)/N)
+   improvement[i] <- (sum(correct)/N - 0.5)/0.5
+   
+ }
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|======================================================================| 100%> close(pb)

> 
> pdf(file="copula-error-rate.pdf", width=5, height=5.5)
> 
> par(mar=c(5,5,4,2))
> 
> scatter.smooth(error.rate ~ rho,
+                xlab = c("correlation between true instructor quality", "and average student evaluation"),
+                #ylab = c("error rate (% of time higher quality instructor", "has lower average evaluation"),
+                ylab= "",
+                span = 0.2, degree=2, col="darkgray"
+ )
> mtext(text = "error rate (% of time higher quality instructor", side=2, line=4)
> mtext(text = "has lower average evaluation)", side=2, line=3)
> 
> clip(-1,.4,-1,error.rate[rho==0.4])
> abline(v=0.4, col="darkgray", lty=2)
> abline(h=error.rate[rho==0.4], col="darkgray", lty=2)
> clip(-1,0.9,-1,error.rate[rho==0.9])
> abline(v=0.9, col="darkgray", lty=2)
> abline(h=error.rate[rho==0.9], col="darkgray", lty=2)
> box()
> 
> dev.off()
pdf 
  2 
> 
> error.rate[rho==0.4]
[1] 36.9582
> error.rate[rho==0.9]
[1] 14.3106
> 
> 
> #################################
> # Figure 3
> #################################
> 
> 
> rm(list=ls())
> 
> library(copula)
> set.seed(123456)
> 
> N <- 500000
> rho <- 0.4
> gap <- seq(from=0, to=1, by=0.005)
> error.rate <- c()
> decision.rate <- c()
> 
> pb <- txtProgressBar(0, length(gap), style=3)
  |                                                                              |                                                                      |   0%> for(i in seq_along(gap)){
+   
+   setTxtProgressBar(pb, i)
+   norm.cop <- normalCopula(rho)
+   draws1 <- rCopula(N, norm.cop)
+   draws2 <- rCopula(N, norm.cop)
+   
+   eval.m <- ifelse(draws1[,1] - draws2[,1] > gap[i], 1,
+                    ifelse( draws2[,1] - draws1[,1] > gap[i], -1, 0))
+   #effect.m <- draws1[,2] > draws2[,2]
+   
+   decision <- eval.m != 0
+   incorrect <- (eval.m == 1 & (draws1[,2] < draws2[,2])) | (eval.m == -1 & (draws1[,2] > draws2[,2]))
+   error.rate[i] <- 100*(sum(incorrect)/N)
+   decision.rate[i] <- 100*(sum(decision)/N)
+   
+ }
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> close(pb)

> 
> gap <- 100*gap
> 
> pdf(file="copula-gap-error.pdf", width=5.5, height=5.85)
> 
> par(mar=c(5,5,4,2))
> scatter.smooth(error.rate ~ gap,
+                xlab = c("minimum diff. in average evaluation scores","(in percentiles) required for decision"),
+                ylab = "",
+                span = 0.2, degree=2, col="darkgray"
+ )
> text(57.5, 37, labels = paste("correlation between avg. evaluation and quality = ", rho, sep=""),
+      cex=0.8)
> mtext(text = "error rate (% of time better teacher has average evaluation",
+       side=2, line=4)
> mtext(text = "lower by more than the percentile gap on the x-axis)",
+       side=2, line=3)
> 
> x1 <- gap[which(abs(10-error.rate) == min(abs(10-error.rate)))]
> x2 <- gap[which(abs(5-error.rate) == min(abs(5-error.rate)))]
> clip(-10,x1,-10,10)
> abline(v=x1, col="darkgray", lty=2)
> abline(h=10, col="darkgray", lty=2)
> clip(-10,x2,-10,5)
> abline(v=x2, col="darkgray", lty=2)
> abline(h=5, col="darkgray", lty=2)
> box()
> 
> 
> dev.off()
pdf 
  2 
> 
> gap[which(abs(10-error.rate) == min(abs(10-error.rate)))]
[1] 39.5
> error.rate[which(abs(10-error.rate) == min(abs(10-error.rate)))]
[1] 10.05
> 
> gap[which(abs(5-error.rate) == min(abs(5-error.rate)))]
[1] 54
> error.rate[which(abs(5-error.rate) == min(abs(5-error.rate)))]
[1] 5.0646
> 
> decision.rate[which(abs(5-error.rate) == min(abs(5-error.rate)))]
[1] 21.0042
> 
> 
> 
> 
> #################################
> # Figure 4
> #################################
> 
> rm(list=ls())
> 
> library(copula)
> set.seed(123456)
> 
> N <- 1000000
> rho <- 0.4
> gap <- 0
> error.rate <- c()
> decision.rate <- c()
> 
> norm.cop <- normalCopula(rho)
> draws1 <- rCopula(N, norm.cop)
> 
> scores <- apply(X=draws1, FUN=quantile, MARGIN=2, probs=seq(0.05, 0.95, 0.05))
> 
> # proportion of faculty with evaluations below 20th percentile
> # who are above-average quality teachers
> 1-ecdf(draws1[draws1[,1]<=scores["20%",1],2])(0.5)
[1] 0.27425
> 
> pdf(file="copula-probability-density.pdf", width=5.5, height=5.85)
> 
> par(mar=c(5,5,4,2))
> 
> y.r <- (draws1[draws1[,1]<=scores["20%",1],2])*100
> hist(y.r, freq=F, breaks=10, ylim=c(0, 0.024), yaxt='n',
+      #main = c("distribution of teacher quality", "student evaluations at or below 20th percentile"),
+      main="", col= c(rep("white", 5), rep("gray", 5)),
+      xlab = "percentile of teacher quality", ylab="")
> axis(2, at=seq(0, 0.024, 0.004), las=1, labels=seq(0, 0.24, 0.04))
> mtext(text="density (proportion of faculty in bin)", side=2, line=4)
> legend("topright", fill=c("white", "gray"), 
+        legend=c("at or below median quality", "above median quality"))
> 
> dev.off()
pdf 
  2 
> 
> 
> # proportion of faculty with evaluations above 80th percentile
> # who are at or below the median in quality
> ecdf(draws1[draws1[,1]>scores["80%",1],2])(0.5)
[1] 0.274715
> 
> # proportion of faculty with evaluations at or below 20th percentile
> # who are above the median in quality
> 1 - ecdf(draws1[draws1[,1]<=scores["20%",1],2])(0.5)
[1] 0.27425
> 
> # proportion of faculty with evaluations above 95th percentile
> # who are at or below the median in quality
> ecdf(draws1[draws1[,1]>scores["95%",1],2])(0.5)
[1] 0.18928
> 
> # proportion of faculty with evaluations at or below 5th percentile
> # who are above the median in quality
> 1 - ecdf(draws1[draws1[,1]<=scores["5%",1],2])(0.5)
[1] 0.1858
> 
> 
> #################################
> # Appendix Figure 7
> #################################
> 
> pdf(file="copula-probability-density-highrate.pdf", width=5.5, height=5.85)
> 
> par(mar=c(5,5,4,2))
> 
> y.r <- (draws1[draws1[,1]>scores["95%",1],2])*100
> hist(y.r, freq=F, breaks=10, ylim=c(0, 0.032), yaxt='n',
+      #main = c("distribution of teacher quality", "student evaluations above 50th percentile"),
+      main="", col= c(rep("white", 5), rep("gray", 5)),
+      xlab = "percentile of teacher quality", ylab="")
> axis(2, at=seq(0, 0.032, 0.004), las=1, labels=seq(0, 0.32, 0.04))
> mtext(text="density (proportion of faculty in bin)", side=2, line=4)
> legend(x=5, y=0.023, fill=c("white", "gray"), 
+        legend=c("at or below median quality", "above median quality"))
> 
> dev.off()
pdf 
  2 
> 
> 
> 
> 
> #################################
> # Appendix Figure 11
> #################################
> 
> rm(list=ls())
> 
> library(mvtnorm)
> set.seed(123456)
> 
> N <- 1000000
> rho <- 0.4
> gap <- 0
> 
> sigma.mat <- matrix(data=c(1, rho, rho, 1), nrow=2, ncol=2, byrow=T)
> draws1 <- rmvnorm(N, mean=c(0,0), sigma=sigma.mat)
> scores <- apply(X=draws1, FUN=quantile, MARGIN=2, probs=seq(0.05, 0.95, 0.05))
> 
> # proportion of faculty with evaluations below 20th percentile
> # who are above-average quality teachers
> 1-ecdf(draws1[draws1[,1]<=scores["20%",1],2])(0)
[1] 0.27425
> 
> pdf(file="probability-density.pdf", width=5.5, height=5.85)
> 
> par(mar=c(5,5,4,2))
> 
> y.r <- (draws1[draws1[,1]<=scores["20%",1],2])
> hist(y.r, freq=F, ylim=c(0, 0.5), yaxt='n',
+      #main = c("distribution of teacher quality", "student evaluations below 20th percentile"),
+      main="", col= c(rep("white", 12), rep("gray", 9)),
+      xlab = "standardized teacher quality metric", ylab="")
> 
> legend("topright", fill=c("white", "gray"), 
+        legend=c("at or below median quality", "above median quality"))
> 
> axis(2, at=seq(0, 0.55, 0.05), las=1, labels=seq(0, 0.55, 0.05))
> mtext(text="probability density", side=2, line=4)
> 
> dev.off()
pdf 
  2 
> 
> 
> # proportion of faculty with evaluations above 80th percentile
> # who are at or below the median in quality
> ecdf(draws1[draws1[,1]>scores["80%",1],2])(0)
[1] 0.274715
> 
> # proportion of faculty with evaluations at or below 20th percentile
> # who are above the median in quality
> 1 - ecdf(draws1[draws1[,1]<=scores["20%",1],2])(0)
[1] 0.27425
> 
> # proportion of faculty with evaluations above 95th percentile
> # who are at or below the median in quality
> ecdf(draws1[draws1[,1]>scores["95%",1],2])(qnorm(0.50))
[1] 0.18928
> 
> # proportion of faculty with evaluations at or below 5th percentile
> # who are above the median in quality
> 1 - ecdf(draws1[draws1[,1]<=scores["5%",1],2])(qnorm(0.50))
[1] 0.1858
> 
> #################################
> # Appendix Figure 12
> #################################
> 
> pdf(file="probability-density-highrate.pdf", width=5.5, height=5.85)
> 
> par(mar=c(5,5,4,2))
> 
> y.r <- (draws1[draws1[,1]>scores["95%",1],2])
> hist(y.r, freq=F, ylim=c(0, 0.55), yaxt='n',
+      #main = c("distribution of teacher quality", "student evaluations below 20th percentile"),
+      main="", col= c(rep("white", 8), rep("gray", 11)),
+      xlab = "standardized teacher quality metric", ylab="")
> 
> legend("topright", fill=c("white", "gray"), 
+        legend=c("at or below median quality", "above median quality"))
> 
> axis(2, at=seq(0, 0.55, 0.05), las=1, labels=seq(0, 0.55, 0.05))
> mtext(text="probability density", side=2, line=4)
> 
> dev.off()
pdf 
  2 
> 
> 
> 
> 
> #################################
> # Figure 6
> #################################
> 
> 
> rm(list=ls())
> 
> library(mvtnorm)
> set.seed(123456)
> 
> draws1 <- rmvnorm(9000, mean=c(0.2,0.2), sigma=matrix(data=c(0.3,0,0,1.3), ncol=2, nrow=2))
> draws2 <- rmvnorm(1000, mean=c(-2,-2), sigma=matrix(data=c(0.1,0,0,0.1), ncol=2, nrow=2))
> draws <- rbind(draws1, draws2)
> 
> cor(draws)
          [,1]      [,2]
[1,] 1.0000000 0.4035215
[2,] 0.4035215 1.0000000
> draws[,1] <- draws[,1] - mean(draws[,1])
> draws[,1] <- draws[,1] / sd(draws[,1])
> cor(draws)
          [,1]      [,2]
[1,] 1.0000000 0.4035215
[2,] 0.4035215 1.0000000
> mean(draws[,1])
[1] 6.766202e-18
> mean(draws[,2])
[1] -0.007020039
> sd(draws[,1])
[1] 1
> sd(draws[,2])
[1] 1.273486
> 
> pdf(file="bimodal-cor.pdf", width=5, height=5.85)
> 
> plot(draws, cex=0.5, col=gray(0.6, alpha=0.25),
+      xlab="standardized SET score", ylab="raw faculty quality metric",
+      xlim=c(-3.5, 3.5), ylim=c(-6.5,6.5))
> q1 <- quantile(ecdf(draws[,1]), 0.1)
> q1
      10% 
-1.550453 
> abline(v=q1, lty=2)
> abline(h=0, lty=2)
> 
> dev.off()
pdf 
  2 
> 
> sel <- draws[,1] < q1
> 1 - mean( draws[sel==1,2] <= 0 )
[1] 0.007
> 
> vcv <- cov(draws)
> 
> draws <- rmvnorm(10000, mean=c(0,0), sigma=vcv)
> cor(draws)
         [,1]     [,2]
[1,] 1.000000 0.401389
[2,] 0.401389 1.000000
> mean(draws[,1])
[1] -0.006006681
> mean(draws[,2])
[1] 0.001697388
> sd(draws[,1])
[1] 1.004167
> sd(draws[,2])
[1] 1.272884
> 
> pdf(file="unimodal-cor.pdf", width=5, height=5.85)
> 
> plot(draws, cex=0.5, col=gray(0.6, alpha=0.25),
+      xlab="standardized SET score", ylab="raw faculty quality metric",
+      xlim=c(-3.5, 3.5), ylim=c(-6.5,6.5))
> q2 <- quantile(ecdf(draws[,1]), 0.1)
> q2
    10% 
-1.2836 
> abline(v=q2, lty=2)
> abline(h=0, lty=2)
> 
> dev.off()
pdf 
  2 
> 
> sel <- draws[,1] < q2
> 1 - mean( draws[sel==1,2] <= 0 )
[1] 0.217
> 
> 
> 
> 
> 
> 
> 
> 
> #################################
> # Figure 5
> #################################
> 
> rm(list=ls())
> 
> library(mvtnorm)
> library(copula)
> library(gplots)

Attaching package: 'gplots'

The following object is masked from 'package:stats':

    lowess

> 
> set.seed(123456)
> mcl.vec <- seq(from = 0, to = 0.9, by = 0.1)
> reps <- 10000
> N <- 1000
> rho <- 0.4
> accuracy.4 <- c()
> ui.4.v <- c()
> li.4.v <- c()
> accuracy.3 <- c()
> ui.3.v <- c()
> li.3.v <- c()
> accuracy.2 <- c()
> ui.2.v <- c()
> li.2.v <- c()
> 
> 
> for(j in 1:length(mcl.vec)){
+  
+   mcl <- mcl.vec[j]
+   cat("Correlation = ", mcl.vec[j], "\n", "\n")
+   
+   cor.store.4 <- c()
+   cor.store.3 <- c()
+   cor.store.2 <- c()
+   
+   pb <- txtProgressBar(min=1, max=reps, initial=0, style=3)
+    
+   for(i in 1:reps){
+     
+     setTxtProgressBar(pb, i)
+   
+     vcv <- diag(5)
+     vcv[1,] <- c(1, rho, rho, rho, rho)
+     vcv[2,] <- c(rho, 1, mcl, mcl, mcl)
+     vcv[3,] <- c(rho, mcl, 1, mcl, mcl)
+     vcv[4,] <- c(rho, mcl, mcl, 1, mcl)
+     vcv[5,] <- c(rho, mcl, mcl, mcl, 1)
+     
+     norm.cop <- normalCopula(P2p(vcv), dim=5, dispstr="un")
+     draws <- rCopula(N, norm.cop)
+     colnames(draws) <- c("true.q", "s1", "s2", "s3", "s4")
+     
+     dat <- data.frame(draws)
+     
+     dat.s <- subset(dat, select=(-true.q))
+     pred.q <- apply(X=dat.s, FUN=mean, MARGIN=1)
+     cor.store.4[i] <- cor(pred.q, dat$true.q)
+     
+     dat.s <- subset(dat, select=(-c(true.q, s4)))
+     pred.q <- apply(X=dat.s, FUN=mean, MARGIN=1)
+     cor.store.3[i] <- cor(pred.q, dat$true.q)
+     
+     dat.s <- subset(dat, select=(-c(true.q, s4, s3)))
+     pred.q <- apply(X=dat.s, FUN=mean, MARGIN=1)
+     cor.store.2[i] <- cor(pred.q, dat$true.q)
+     
+     
+   
+   }
+   
+   accuracy.4[j] <- mean(cor.store.4) 
+   ui.4.v[j] <- quantile(cor.store.4, probs=0.975)
+   li.4.v[j] <- quantile(cor.store.4, probs=0.025)
+   
+   accuracy.3[j] <- mean(cor.store.3) 
+   ui.3.v[j] <- quantile(cor.store.3, probs=0.975)
+   li.3.v[j] <- quantile(cor.store.3, probs=0.025)
+   
+   accuracy.2[j] <- mean(cor.store.2) 
+   ui.2.v[j] <- quantile(cor.store.2, probs=0.975)
+   li.2.v[j] <- quantile(cor.store.2, probs=0.025)
+   
+   close(pb)
+   
+ }
Correlation =  0 
 
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Correlation =  0.1 
 
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Correlation =  0.2 
 
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Correlation =  0.3 
 
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Correlation =  0.4 
 
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Correlation =  0.5 
 
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Correlation =  0.6 
 
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Correlation =  0.7 
 
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Correlation =  0.8 
 
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Correlation =  0.9 
 
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> 
> plotCI(x = mcl.vec+0.03, y = accuracy.4, ui=ui.4.v, li=li.4.v,
+        xlab = "collinearity among noisy measures",
+        ylab = "correlation of average with quality", 
+        xlim=c(-0.05, 1.05), ylim=c(0.3, 0.85), sfrac=0.005)
> 
> plotCI(x = mcl.vec, y = accuracy.3, ui=ui.3.v, li=li.3.v,
+        add=T, sfrac=0.005)
> 
> plotCI(x = mcl.vec-0.03, y = accuracy.2, ui=ui.2.v, li=li.2.v,
+        add=T, sfrac=0.005)
> 
> pdf(file="averaged-measure.pdf", width=5, height=5.85)
> 
> plot(accuracy.4 ~ mcl.vec,
+      xlab = "collinearity among noisy measures",
+      ylab = "correlation between averaged measure and instructor quality", 
+      xlim=c(-0.05, 1.05), ylim=c(0.38, 0.8), type="l")
> #lines(ui.4.v ~ mcl.vec, lty=2)
> #lines(li.4.v ~ mcl.vec, lty=2)
> 
> lines(accuracy.3 ~ mcl.vec, col=gray(0.6), lty=2)
> #lines(ui.3.v ~ mcl.vec, lty=2, col=gray(0.7))
> #lines(li.3.v ~ mcl.vec, lty=2, col=gray(0.7))
> 
> lines(accuracy.2 ~ mcl.vec, col=gray(0.8), lwd=2)
> #lines(ui.2.v ~ mcl.vec, lty=2, col=gray(0.6))
> #lines(li.2.v ~ mcl.vec, lty=2, col=gray(0.6))
> 
> legend("topright", lty=c(1,2,1), lwd=c(1,1,2), col=c("black", gray(0.6), gray(0.8)), legend = c("4 measures", "3 measures", "2 measures"))
> 
> dev.off()
pdf 
  2 
> 
> 
> 
> 
> 
> 
> #################################
> # Appendix Figure 13
> #################################
> 
> rm(list=ls())
> 
> library(mvtnorm)
> library(copula)
> library(gplots)
> 
> set.seed(123456)
> mcl.vec <- seq(from = 0, to = 0.9, by = 0.1)
> reps <- 10000
> N <- 1000
> rho <- 0.4
> accuracy.4 <- c()
> ui.4.v <- c()
> li.4.v <- c()
> accuracy.3 <- c()
> ui.3.v <- c()
> li.3.v <- c()
> accuracy.2 <- c()
> ui.2.v <- c()
> li.2.v <- c()
> 
> 
> for(j in 1:length(mcl.vec)){
+   
+   mcl <- mcl.vec[j]
+   cat("Correlation = ", mcl.vec[j], "\n", "\n")
+   
+   cor.store.4 <- c()
+   cor.store.3 <- c()
+   cor.store.2 <- c()
+   
+   pb <- txtProgressBar(min=1, max=reps, initial=0, style=3)
+   
+   for(i in 1:reps){
+     
+     setTxtProgressBar(pb, i)
+     
+     vcv <- diag(5)
+     vcv[1,] <- c(1, rho, rho, rho, rho)
+     vcv[2,] <- c(rho, 1, mcl, mcl, mcl)
+     vcv[3,] <- c(rho, mcl, 1, mcl, mcl)
+     vcv[4,] <- c(rho, mcl, mcl, 1, mcl)
+     vcv[5,] <- c(rho, mcl, mcl, mcl, 1)
+     
+     #norm.cop <- normalCopula(P2p(vcv), dim=5, dispstr="un")
+     #draws <- rCopula(N, norm.cop)
+     draws <- rmvnorm(N, mean=rep(0,5), sigma=vcv)
+     colnames(draws) <- c("true.q", "s1", "s2", "s3", "s4")
+     
+     dat <- data.frame(draws)
+     
+     dat.s <- subset(dat, select=(-true.q))
+     pred.q <- apply(X=dat.s, FUN=mean, MARGIN=1)
+     cor.store.4[i] <- cor(pred.q, dat$true.q)
+     
+     dat.s <- subset(dat, select=(-c(true.q, s4)))
+     pred.q <- apply(X=dat.s, FUN=mean, MARGIN=1)
+     cor.store.3[i] <- cor(pred.q, dat$true.q)
+     
+     dat.s <- subset(dat, select=(-c(true.q, s4, s3)))
+     pred.q <- apply(X=dat.s, FUN=mean, MARGIN=1)
+     cor.store.2[i] <- cor(pred.q, dat$true.q)
+     
+     
+     
+   }
+   
+   accuracy.4[j] <- mean(cor.store.4) 
+   ui.4.v[j] <- quantile(cor.store.4, probs=0.975)
+   li.4.v[j] <- quantile(cor.store.4, probs=0.025)
+   
+   accuracy.3[j] <- mean(cor.store.3) 
+   ui.3.v[j] <- quantile(cor.store.3, probs=0.975)
+   li.3.v[j] <- quantile(cor.store.3, probs=0.025)
+   
+   accuracy.2[j] <- mean(cor.store.2) 
+   ui.2.v[j] <- quantile(cor.store.2, probs=0.975)
+   li.2.v[j] <- quantile(cor.store.2, probs=0.025)
+   
+   close(pb)
+   
+ }
Correlation =  0 
 
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Correlation =  0.1 
 
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Correlation =  0.2 
 
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Correlation =  0.3 
 
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Correlation =  0.4 
 
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Correlation =  0.5 
 
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Correlation =  0.6 
 
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Correlation =  0.7 
 
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Correlation =  0.8 
 
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Correlation =  0.9 
 
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> 
> plotCI(x = mcl.vec+0.03, y = accuracy.4, ui=ui.4.v, li=li.4.v,
+        xlab = "collinearity among noisy measures",
+        ylab = "correlation of average with quality", 
+        xlim=c(-0.05, 1.05), ylim=c(0.3, 0.85), sfrac=0.005)
> 
> plotCI(x = mcl.vec, y = accuracy.3, ui=ui.3.v, li=li.3.v,
+        add=T, sfrac=0.005)
> 
> plotCI(x = mcl.vec-0.03, y = accuracy.2, ui=ui.2.v, li=li.2.v,
+        add=T, sfrac=0.005)
> 
> pdf(file="averaged-measure-normal.pdf", width=5, height=5.85)
> 
> plot(accuracy.4 ~ mcl.vec,
+      xlab = "collinearity among noisy measures",
+      ylab = "correlation between averaged measure and instructor quality", 
+      xlim=c(-0.05, 1.05), ylim=c(0.38, 0.8), type="l")
> #lines(ui.4.v ~ mcl.vec, lty=2)
> #lines(li.4.v ~ mcl.vec, lty=2)
> 
> lines(accuracy.3 ~ mcl.vec, col=gray(0.6), lty=2)
> #lines(ui.3.v ~ mcl.vec, lty=2, col=gray(0.7))
> #lines(li.3.v ~ mcl.vec, lty=2, col=gray(0.7))
> 
> lines(accuracy.2 ~ mcl.vec, col=gray(0.8), lwd=2)
> #lines(ui.2.v ~ mcl.vec, lty=2, col=gray(0.6))
> #lines(li.2.v ~ mcl.vec, lty=2, col=gray(0.6))
> 
> legend("topright", lty=c(1,2,1), lwd=c(1,1,2), col=c("black", gray(0.6), gray(0.8)), legend = c("4 measures", "3 measures", "2 measures"))
> 
> dev.off()
pdf 
  2 
> 
> 
> proc.time()
   user  system elapsed 
7303.39   37.23 3781.67 
